Question

A fraction becomes 1/2 when 1 is subtracted from its numerator and 1 is added to its denominator. Also, it becomes 1/3 when 6 is subtracted from its numerator and 1 from the denominator, Find original fraction

Answer 1

Given:

A fraction becomes 1/2 when 1 is subtracted from its numerator and 1 is added to its denominator

Also, it becomes 1/3 when 6 is subtracted from its numerator and 1 from the denominator

To find:

The original

Solution:

Let,

"x" be the numerator

"y" be the denominator

So, the original fraction =  $\frac{x}{y}$

(1) When 1 is subtracted from the numerator and 1 is added to its denominator, we get:

$\frac{x - 1}{y + 1} = \frac{1}{2}$

$\implies 2(x - 1) = y + 1$

$\implies 2x - 2 = y + 1$

$\implies 2x - y -3 = 0$ ...... (i)

(2) When 6 is subtracted from the numerator and 1 is subtracted from its denominator, we get:

$\frac{x - 6}{y - 1} = \frac{1}{3}$

$\implies 3(x - 6) = y - 1$

$\implies 3x - 18 = y - 1$

$\implies 3x - y -17 = 0$ ...... (ii)

Now, Subtracting eq. (i) from (ii), we get

3x - y - 17 = 0

2x - y - 3 = 0

-    +    +

---------------------

x -14 = 0

----------------------

∴ x = 14

Substituting x = 14 in eq. (i), we get

2(14) - y - 3 = 0

⇒ 28 - y - 3 = 0

⇒ 25 - y = 0

⇒ y = 25

Thus, $\boxed{\bold{The \:original\:fraction\:is\:\:\underline{\frac{14}{25}} }}\:.$

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Answer 2

Step-by-step explanation:

Given:

A fraction becomes 1/2 when 1 is subtracted from its numerator and 1 is added to its denominator

Also, it becomes 1/3 when 6 is subtracted from its numerator and 1 from the denominator

To find:

The original

Solution:

Let,

"x" be the numerator

"y" be the denominator

So, the original fraction = xy\frac{x}{y}

y

x

(1) When 1 is subtracted from the numerator and 1 is added to its denominator, we get:

x−1y+1=12\frac{x - 1}{y + 1} = \frac{1}{2}

y+1

x−1

=

2

1

⟹2(x−1)=y+1\implies 2(x - 1) = y + 1⟹2(x−1)=y+1

⟹2x−2=y+1\implies 2x - 2 = y + 1⟹2x−2=y+1

⟹2x−y−3=0\implies 2x - y -3 = 0⟹2x−y−3=0 ...... (i)

(2) When 6 is subtracted from the numerator and 1 is subtracted from its denominator, we get:

x−6y−1=13\frac{x - 6}{y - 1} = \frac{1}{3}

y−1

x−6

=

3

1

⟹3(x−6)=y−1\implies 3(x - 6) = y - 1⟹3(x−6)=y−1

⟹3x−18=y−1\implies 3x - 18 = y - 1⟹3x−18=y−1

⟹3x−y−17=0\implies 3x - y -17 = 0⟹3x−y−17=0 ...... (ii)

Now, Subtracting eq. (i) from (ii), we get

3x - y - 17 = 0

2x - y - 3 = 0

- + +

---------------------

x -14 = 0

----------------------

∴ x = 14

Substituting x = 14 in eq. (i), we get

2(14) - y - 3 = 0

⇒ 28 - y - 3 = 0

⇒ 25 - y = 0

⇒ y = 25

Thus, Theoriginalfractionis1425‾.\boxed{\bold{The \:original\:fraction\:is\:\:\underline{\frac{14}{25}} }}\:.

Theoriginalfractionis

25

14

.

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